the Fock matrix can be re-block-diagonalized by the unitary transformation[1]}
\(\mathbf {U}&=&\mathbf {C}\mathbf {T},\\\mathbf {T}&=&\begin{pmatrix}\mathbf {C}_{\mathrm {cc}}^\dag (\mathbf {C}_{\mathrm {cc}}\mathbf {C}_{\mathrm {cc}}^\dag )^{-1/2}&\mathbf {0}&\mathbf {0}\\\mathbf {0}&\mathbf {C}_{\mathrm {aa}}^\dag (\mathbf {C}_{\mathrm {aa}}\mathbf {C}_{\mathrm {aa}}^\dag )^{-1/2}&\mathbf {0}\\\mathbf {0}&\mathbf {0}&\mathbf {C}_{\mathrm {vv}}^\dag (\mathbf {C}_{\mathrm {vv}}\mathbf {C}_{\mathrm {vv}}^\dag )^{-1/2}\end{pmatrix},\) 
